- Tab to move across fields.
- Left-to-right, top-to-bottom focus.
- Space to toggle togglable stuff.
- Alt-Down Arrow to deploy drop-down stuff.
- Arrows to move around.
- Enter/Esc to accept/discard a modal.
Or maybe I misunderstand the things you call "mouse things".
Keyboard design is much more, its also about how and where to place items, how things should move... it's a whole world. In the mouse world we have "discovered" those things and now we "just copy" other designs that we know work fine.
If this grid a central component of the application and navigation inside it is very common, I would argue that there should be a way to quickly move around the cells. The "how" is up to the specific application and the paradigms it promotes. For example, line navigation in vim is possible with "<N>G", but such a shortcut would seem absurd in another application.
- Spreadsheet-like stuff: arrows, PgDn, PgUp, Home, End, … move inside of it.